Riemannian curvature

Riemannian curvature

[rē′män·ē·ən ′kər·və·chər]
(mathematics)
A general notion of space curvature at a point of a Riemann space which is directly obtained from orthonormal tangent vectors there.
References in periodicals archive ?
R, S, Q & r being Christoffel Riemannian curvature tensor, Ricci tensor, Ricci operator and scalar curvature respectively.
[OMEGA] = ([[OMEGA].sup.i.sub.j]) is called curvature 2-form or the Riemannian curvature tensor associated with the connection [theta] and it is written as
for any X = ([X.sub.1], [X.sub.2]), Y = ([Y.sub.1], [Y.sub.2]), Z = ([Z.sub.1], [Z.sub.2]) [member of] T([M.sub.1] x [M.sub.2]), where R, [R.sub.1] and [R.sub.2] are respectively the Riemannian curvature tensors of the Riemannian manifolds ([M.sub.1] x [M.sub.2], g), ([M.sub.1], [g.sub.1]) and ([M.sub.2], [g.sub.2]);
The advantage of the current theory proposed in this paper is that it provides an analytical approach based on Riemannian curvature, and the dynamics of gravitation mathematically represented by differential gravity calculations around the points of constant curvature.
where [R.sup.N] and {[e.sub.i]} are the Riemannian curvature of N , and a local orthonormal frame field of M , respectively, (EELLS; SAMPSON, 1964), (JIANG, 1986a, 1986b), (TURHAN; KORPINAR, 2010).
(1) C x [sub.f] N has harmonic Riemannian curvature tensor;
Among the topics are calculus and heat flow in metric measure spaces and spaces with Riemannian curvature bounded from below, Ma-Trudinger-Wang curvature and regularity of optimal transport maps, and a proof of Bobkov's spectral bound for convex domains via Gaussian fitting and free energy estimation.
Let M = [M.sub.1] x [sub.f][M.sub.2] be a warped product and R and [??] denote the Riemannian curvature tensors of M with respect to the Levi-Civita connection and the semi-symmetric non-metric connection, respectively.
Since the Riemannian curvature tensor [??] on [M.sup.n] is given by (see (1.1)
We denote by M(c), then the Riemannian curvature tensor of M(c) is given by
Over the course of five chapters he discusses the geometry of the Riemannian curvature tensor, curvature homogeneous generalized plane wave manifolds, examples which are not generalized plane wave manifolds, the algebraics of the curvature tensor, complex models that are both Osserman and complex Osserman, and introductory Stanilov-Tsankov theory.
If M is a manifold, we look for a complete time-dependent metric on M with bounded curvature, measuring a smooth one-parameter family of metrics G = G(t) for A [less than] t [less than] [Omega] where each G(t) = {[g.sub.ij](x, t)[dx.sup.i][dx.sup.j]} is a complete metric with Riemannian curvature bounded by a constant B independent of t.